Question

A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height to the radius of the cone so formed?

• A. 4 : 3
• B. 2 : 3
• C. 3 : 4
• D. None of these

Hint:

Equal volumes with the same radius give h = 4r, a 4:1 ratio, which isn't among the first three choices.

Answer 1

The Correct Option is D

Step 1: Set up the volumes.
Let the sphere's radius be r. Volume of sphere \(=\dfrac{4}{3}\pi r^3\). The cone has the same radius r and some height h, so its volume \(=\dfrac{1}{3}\pi r^2h\).

Step 2: Equate the volumes (same material, recast).
\(\dfrac{4}{3}\pi r^3=\dfrac{1}{3}\pi r^2h\).

Step 3: Solve for h in terms of r.
Cancelling \(\dfrac{1}{3}\pi r^2\) from both sides: \(4r=h\).

Step 4: Find the ratio.
\(\dfrac{h}{r}=\dfrac{4r}{r}=4\), i.e. the ratio of height to radius is 4:1.

Step 5: Compare with the options.
4:1 does not match 4:3, 2:3, or 3:4, so none of the specific ratios given is correct.

Step 6: Final Answer.
The ratio is 4:1, which is not listed, so option D, none of these, is correct.